Christophe LETELLIER
31/12/2010

**Christian Mira**

In the early 70s, Igor Gumowski and Christian Mira studied families of maps obtained from a dissipative perturbation of conservative maps issued from the studies on the problem of ``stochastic’’ instability in accelerators and storage rings [1]. Among these two families, there is a two-dimensional quasi-conservative map which may be written as

where parameters values were and , being the bifurcation parameter. This two-dimensional map was published in 1973 [2] and republished in 1980 [3]. Two chaotic attractors were presented as being ``stochastic’’ in 1973 (Fig. 1).

[1] C. Mira, Memories of the early days of chaos theory in *The Chaos avant-garde*, *World Scientific Series on Nonlinear Science A*, **39**, pp.
95-197, 2000.

[2] J. Bernussen, Liu Hsu & C. Mira, Quelques exemples du second ordre,
Collected preprints of Colloque International du CNRS, **229**, *Transformations ponctuelles et applications* (Toulouse, September 1973),
Proceedings Editions du CNRS (Paris), pp. 195-226, 1976.

[3] I. Gumowski & C. Mira, Recurrences and discrete dynamic systems : An introduction, *Lecture Notes in Mathematics*, **809**, 1980.